--- a/equalizer.agda	Sun Sep 01 22:10:39 2013 +0900
+++ b/equalizer.agda	Mon Sep 02 00:12:32 2013 +0900
@@ -21,10 +21,22 @@
 open import HomReasoning
 open import cat-utility
 
-record Equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ }  {a b : Obj A} (f g : Hom A a b) : Set  ( ℓ ⊔ (c₁ ⊔ c₂)) where
+record Equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ }  {a b : Obj A} (f g : Hom A a b) : Set  (ℓ ⊔ (c₁ ⊔ c₂)) where
    field
       equalizer : {c d : Obj A} (f' : Hom  A c a) (g' : Hom A d a) →  Hom A c d 
       equalize : {c d : Obj A} (f' : Hom  A c a) (g' : Hom A d a) →
            A [ A [ f  o  f' ]  ≈ A [  A [ g  o g' ] o equalizer f' g' ] ]
       uniqueness : {c d : Obj A} (f' : Hom  A c a) (g' : Hom A d a) ( e : Hom A c d ) → 
            A [ A [ f  o  f' ]  ≈ A [  A [ g  o g' ] o e ] ] → A [ e  ≈ equalizer f' g' ]
+
+record EqEqualizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ }  {a b : Obj A} (f g : Hom A a b) : Set  (ℓ ⊔ (c₁ ⊔ c₂)) where
+   field
+      α : {d a b : Obj A}  → (f : Hom A a b) → (g : Hom A a b ) →  Hom A d a
+      γ : {d c : Obj A}  → (f : Hom A c b) → (g : Hom A c b ) → (h : Hom A d c ) →  Hom A d c 
+      δ : {a b : Obj A}  → (f : Hom A a b) → Hom A a a 
+      β : {c a b d : Obj A}  → (f : Hom A a b) → (g : Hom A a b ) →  (h : Hom A c d ) → Hom A c a
+      b1 :  {c : Obj A} → A [ A [ f  o α {c} f g ] ≈ A [ g  o α {c} f g ] ]
+      b2 :  {c d : Obj A } → {h : Hom A d a } → A [ A [ α f g o γ f g h ] ≈ A [ h  o α  (A [ f o h ]) (A [ g o h ]) ] ]
+      b3 :  A [ A [ α f f o δ f ] ≈ id1 A a ]
+      b4 :  {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o  k ] ) ≈ k ]
+      b5 :  {c d : Obj A } → {h : Hom A d a } → A [ β f g h ≈  A [ γ f g h o δ (A [ f o h ]) ] ]